Question: Simplify the following expression: $y = \dfrac{-5x^2+4x+12}{-5x - 6}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-5)}{(12)} &=& -60 \\ {a} + {b} &=& &=& {4} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-60$ and add them together. Remember, since $-60$ is negative, one of the factors must be negative. The factors that add up to ${4}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-6}$ and ${b}$ is ${10}$ $ \begin{eqnarray} {ab} &=& ({-6})({10}) &=& -60 \\ {a} + {b} &=& {-6} + {10} &=& 4 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-5}x^2 {-6}x) + ({10}x +{12}) $ Factor out the common factors: $ x(-5x - 6) - 2(-5x - 6)$ Now factor out $(-5x - 6)$ $ (-5x - 6)(x - 2)$ The original expression can therefore be written: $ \dfrac{(-5x - 6)(x - 2)}{-5x - 6}$ We are dividing by $-5x - 6$ , so $-5x - 6 \neq 0$ Therefore, $x \neq -\frac{6}{5}$ This leaves us with $x - 2; x \neq -\frac{6}{5}$.